Finite Group Databasebeta

← #391372

Subgroups — #391372

Order 512.

910 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2095C211yesmaximal, characteristic
2#14276C211yesmaximal, characteristic
3#14261C211yesmaximal, characteristic
4#11373C211yesmaximal, characteristic
5#11268C211yesmaximal, characteristic
6#11219C212yesmaximal
7#11219C212yesmaximal
8#11190C212yesmaximal
9#11067C212yesmaximal
10#11052C211yesmaximal, characteristic
11#10796C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×#103C2212yes—
14C22×#103C2212yes—
15C2×#391C2211yescharacteristic
16C2×#391C2211yescharacteristic
17C2×#391C2211yescharacteristic
18C2×#389C2212yes—
19C2×#389C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#376C2211yescharacteristic
23#2095C418yescomplemented
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2211yescharacteristic
30C2×#336C2212yes—
31C2×#336C2212yes—
32C2×#336C2211yescharacteristic
33#1443—22no—
34#1442—22no—
35#1441—22no—
36#1441—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2212yes—
40C23×Q8C2311yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—22no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2311yescharacteristic
51C22×C42C2311yescharacteristic
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C2×#107—42no—
55C2×#107—42no—
56C2×#107C2×C418yes—
57C2×#103—42no—
58C2×#103—42no—
59C2×#103—42no—
60C2×#103—42no—
61C2×#103—42no—
62C2×#103C2×C418yes—
63#391—22no—
64#391—42no—
65#391—42no—
66#391C2×C414yes—
67#389—42no—
68#389—42no—
69#389—42no—
70#384—22no—
71#384C2×C414yes—
72#382—42no—
73#376—42no—
74C2×#95—22no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—21no—
78C2×#95—22no—
79C2×#95—21no—
80C2×#95—21no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95D412yes—
86C2×#95D412yes—
87#340—42no—
88#340—42no—
89#340—42no—
90#340—42no—
91#340—42no—
92#336—22no—
93#336—42no—
94#336—42no—
95#336C2×C414yes—
96C22×Q8—21no—
97C22×Q8—22no—
98C22×Q8—22no—
99C22×Q8—22no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×Q8—22no—
103C22×Q8—21no—
104C22×Q8—21no—
105C22×Q8C22×C412yes—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—21no—
112C23×C4C2×D412yes—
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D412yes—
116C23×C4C2411yescharacteristic
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8C22×C414yes—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4—82no—
158C2×C4⋊C4—82no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4C22×C414yes—
162C2×C4⋊C4C22×C412yes—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—21no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42C22×C412yes—
174C2×C42#4112yes—
175#107—48no—
176#103—48no—
177#103—48no—
178#95—28no—
179#95—42no—
180#95—42no—
181#95—44no—
182#95—42no—
183#95—44no—
184#95—44no—
185#95—42no—
186#95—42no—
187#95—82no—
188#95—82no—
189#95—82no—
190#95—82no—
191C24C22×D411yescharacteristic
192C2×Q8—24no—
193C2×Q8—22no—
194C2×Q8—24no—
195C2×Q8—22no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—42no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4C4×D414yes—
268C22×C4C23×C411yescharacteristic
269C22×C4C2×#4111yescharacteristic
270C22×C4H(5,𝔽2)12yes—
271C22×C4H(5,𝔽2)12yes—
272C22×C4H(5,𝔽2)12yes—
273C22×C4H(5,𝔽2)11yescharacteristic
274C22×C42−1+411yescharacteristic
275C22×C42−1+411yescharacteristic
276C22×C42−1+412yes—
277C22×C42−1+411yescharacteristic
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—88no—
307C4⋊C4—84no—
308C4⋊C4—88no—
309C42—24no—
310C42—24no—
311C42—42no—
312C42—42no—
313C42—42no—
314C42—42no—
315C23C2×C4×D411yescharacteristic
316C23#53411yescharacteristic
317C23#53511yescharacteristic
318C23#53511yescharacteristic
319C23#53611yescharacteristic
320C23#53611yescharacteristic
321C23#53711yescharacteristic
322C23#57612yes—
323C23#57712yes—
324C23#57812yes—
325C23#57812yes—
326Q8—44no—
327Q8—44no—
328Q8—82no—
329Q8—82no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—82no—
367C2×C4—81no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4#51812yes—
391C2×C4#51911yescharacteristic
392C2×C4#52011yescharacteristic
393C2×C4#52012yes—
394C2×C4#53912yes—
395C2×C4#54012yes—
396C2×C4#54111yescharacteristic
397C2×C4#54211yescharacteristic
398C2×C4#56012yes—
399C2×C4#56212yes—
400C2×C4#56212yes—
401C2×C4#56212yes—
402C2×C4#56212yes—
403C2×C4#56312yes—
404C2×C4#56312yes—
405C2×C4#56312yes—
406C8—44no—
407C8—82no—
408C22#232211yescharacteristic
409C22#232311yescharacteristic
410C22#232811yescharacteristic
411C22#281311yescharacteristic
412C22#281611yescharacteristic
413C22#282011yescharacteristic
414C22#282111yescharacteristic
415C22#294612yes—
416C22#295012yes—
417C22#320112yes—
418C22#320312yes—
419C22#322011yescharacteristic
420C22#322111yescharacteristic
421C22#322311yescharacteristic
422C22#322311yescharacteristic
423C22#322312yes—
424C22#322412yes—
425C22#324212yes—
426C22#324212yes—
427C22#324312yes—
428C22#324312yes—
429C22#325812yes—
430C22#326212yes—
431C4—44no—
432C4—84no—
433C4—82no—
434C4—81no—
435C4—81no—
436C4—82no—
437C4—82no—
438C4—161no—
439C4—161no—
440C4—162no—
441C4#237612yes—
442C4#237812yes—
443C4#283612yes—
444C4#283712yes—
445C2#1462811yescharacteristic
446C2#2208712yes—
447C2#2212512yes—
448C2#2227711yescharacteristic
449C2#2227911yescharacteristic
450C2#3273612yes—
451C2#3278311yescharacteristic
452C2#3278411yescharacteristic
453C2#3282911yescharacteristic
454C2#3283211yescharacteristic
455C2#3284612yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
23. #2095C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#20952
#142762
#142612
#113732
#112682
#112192
#111902
#110672
#110522
#107962